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Properties

Label 2-1380-92.91-c1-0-48
Degree $2$
Conductor $1380$
Sign $0.915 - 0.401i$
Analytic cond. $11.0193$
Root an. cond. $3.31954$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.433 + 1.34i)2-s − i·3-s + (−1.62 − 1.16i)4-s + i·5-s + (1.34 + 0.433i)6-s + 3.66·7-s + (2.27 − 1.67i)8-s − 9-s + (−1.34 − 0.433i)10-s − 2.17·11-s + (−1.16 + 1.62i)12-s + 6.11·13-s + (−1.58 + 4.92i)14-s + 15-s + (1.27 + 3.79i)16-s − 2.05i·17-s + ⋯
L(s)  = 1  + (−0.306 + 0.951i)2-s − 0.577i·3-s + (−0.811 − 0.583i)4-s + 0.447i·5-s + (0.549 + 0.177i)6-s + 1.38·7-s + (0.804 − 0.593i)8-s − 0.333·9-s + (−0.425 − 0.137i)10-s − 0.656·11-s + (−0.337 + 0.468i)12-s + 1.69·13-s + (−0.424 + 1.31i)14-s + 0.258·15-s + (0.318 + 0.948i)16-s − 0.499i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 - 0.401i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.915 - 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.915 - 0.401i$
Analytic conductor: \(11.0193\)
Root analytic conductor: \(3.31954\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1380,\ (\ :1/2),\ 0.915 - 0.401i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.570470720\)
\(L(\frac12)\) \(\approx\) \(1.570470720\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.433 - 1.34i)T \)
3 \( 1 + iT \)
5 \( 1 - iT \)
23 \( 1 + (-2.44 + 4.12i)T \)
good7 \( 1 - 3.66T + 7T^{2} \)
11 \( 1 + 2.17T + 11T^{2} \)
13 \( 1 - 6.11T + 13T^{2} \)
17 \( 1 + 2.05iT - 17T^{2} \)
19 \( 1 + 1.02T + 19T^{2} \)
29 \( 1 + 6.97T + 29T^{2} \)
31 \( 1 + 3.10iT - 31T^{2} \)
37 \( 1 - 8.95iT - 37T^{2} \)
41 \( 1 - 4.62T + 41T^{2} \)
43 \( 1 - 8.16T + 43T^{2} \)
47 \( 1 + 9.68iT - 47T^{2} \)
53 \( 1 + 5.81iT - 53T^{2} \)
59 \( 1 - 1.80iT - 59T^{2} \)
61 \( 1 - 8.11iT - 61T^{2} \)
67 \( 1 - 15.0T + 67T^{2} \)
71 \( 1 + 4.61iT - 71T^{2} \)
73 \( 1 + 4.66T + 73T^{2} \)
79 \( 1 - 12.9T + 79T^{2} \)
83 \( 1 - 10.8T + 83T^{2} \)
89 \( 1 + 4.31iT - 89T^{2} \)
97 \( 1 - 18.8iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.328422108478493395155773029659, −8.466178176870085734444201215472, −8.041843875585897062905277329310, −7.28617097522128181156287320671, −6.44325932280192354874698312718, −5.65123281898725804543841192621, −4.84667214034831778415477287420, −3.75661005879827910638020366058, −2.16011147176827228236518008985, −0.939333527330057438111314688558, 1.12119335583372253205936393136, 2.12475796230569642784646650866, 3.54097015089514508655498845139, 4.22001193503425303779019354305, 5.13961959356754749990501109036, 5.83486616756761953165595682018, 7.61480144227488051556392979904, 8.116849279722144168766249054784, 8.923478805402364057309453944719, 9.392558495810084497429768557266

Graph of the $Z$-function along the critical line